Stabilité des polynômes

An iterative construction of irreducible polynomials reducible modulo every prime

“…For each m ∈ O it is possible to find infinitely many values of s satisfying the hypotheses of Corollary 3.2. Indeed, fix a prime r of O not dividing (2) or (f n−1 0 (0)), and let x ∈ r/r 2 . By the Chinese remainder theorem there exist infinitely many a ∈ O with a ≡ f n−1 0 (0) + x mod r 2 and a ≡ f n−1 0 (0) mod q.…”

Settled polynomials over finite fields

“…There are some results in special cases, such as f quadratic ( [1], [2], [3], [10], [9]) and f (x) = x m − b [6]. In [9] the first author conjectures that all quadratic f ∈ Z[x] with 0 not periodic are eventually stable (considered as polynomials over Q).…”

“…Previous work on the irreducibility of polynomial iterates (also called stability; see Section 2) has focused principally on the case of polynomials with coefficients in number fields. See for instance [1], [2], [3], [6], [7], [9], and [10]. Settledness was mentioned briefly in Sections 3 and 5 of [4].…”

On the factorization of iterated polynomials

“…The polynomial f (n) (x) is called the n-th iterate of f . Algebraic aspects of f (n) (x) have been extensively considered by many authors in the past few years [1,2,7,8,9]; in most of the cases, the main object of study is the class of polynomials f ∈ K [x] for which its iterates f (n) (x) are all irreducible over K. Such polynomials are called stable. In the case K a finite field, a recent work [6] extends the notion of stability to finite sets of polynomials {f 1 , .…”